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Journal of Mathematical Physics

Coherent states on the m-sheeted covering group of SU(1,1)

Apostol Vourdas

Published Mar 1, 1993• 18 citations
Publisher

Abstract

Coherent states associated with the m-sheeted covering group of SU(1,1) are considered and such properties as overcompleteness and resolution of the identity are studied. They are used to define analytic representations in which the transformations of the m-sheeted covering group of SU(1,1) are implemented as extended Mobius conformal mappings. Applications of this formalism in the study of Hamiltonians that describe m-particle clustering are also considered.

DOI
10.1063/1.530197

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Coherent states associated with the m-sheeted covering group of SU(1,1) are considered and such properties as overcompleteness and resolution of the identity are studied. They are used to define analytic representations in which the transformations of the m-sheeted covering group of SU(1,1) are implemented as extended Mobius conformal mappings. Applications of this formalism in the study of Hamiltonians that describe m-particle clustering are also considered.
SU(2) coherent states on an m-sheeted covering of the sphere are introduced, and properties like overcompleteness and resolution of the identity are studied. Operators J+(m), J−(m), Jz(m) that obey the SU(2) algebra but shift the ‖jn〉 states by m steps are considered, and it is shown that the properties of our coherent states with respect to them are analogous to the properties of the standard SU(2) coherent states with respect to the usual angular momentum operators J+, J−, Jz. An extended SU(2) Bargmann representation on an m-sheeted sphere is introduced in which the transformations of an m-sheeted covering of the SU(2) group are implemented as extended Mobius conformal mappings. The formalism is applied in the study of Hamiltonians that contain the operators J+(m), J−(m), Jz(m) and describe processes where the state of a particle is shifted by m steps. The method is also used in the context of both the Holstein–Primakoff and Schwinger representation of SU(2).
Coherent states on the m-sheeted complex plane are introduced and properties like overcompleteness and resolution of the identity are studied. Operators am+, am which create and annihilate clusters of m particles are considered and it is shown that the properties of our coherent states with respect to them are analogous to the properties of the ordinary (Glauber) coherent states with respect to a+, a. An extended Bargmann representation in the m-sheeted complex plane and P and Q representations with respect to our coherent states, are studied. Applications of this formalism in the study of Hamiltonians that describe m-particle clustering are also considered.
We consider two analytic representations of the SU(1,1) Lie group: the representation in the unit disc based on the SU(1,1) Perelomov coherent states and the Barut - Girardello representation based on the eigenstates of the SU(1,1) lowering generator. We show that these representations are related through a Laplace transform. A `weak' resolution of the identity in terms of the Perelomov SU(1,1) coherent states is presented which is valid even when the Bargmann index k is smaller than . Various applications of these results in the context of the two-photon realization of SU(1,1) in quantum optics are also discussed.